The 2013 article is careful to define what is meant by
random in the modern synthesis:

“I will use the definition that
the changes are assumed to be random with respect to
physiological function and could not therefore be influenced
by such function or by functional changes in response to the
environment. This is the assumption that excludes the
phenotype from in any way influencing or guiding genetic
change.”

Some have criticised
the article and the videos on the grounds that the modern
synthesis does not require DNA change to be equally likely
everywhere in the genome. The article does not make that
claim. On the contrary it states:

“The concept of a purely random
event is not easy to define. The physico-chemical nature of
biological molecules will, in any case, ensure that some
changes are more likely to happen than others.”

Of course the modern
synthesis can accept non-random change. That has been known
ever since the identification of genome hotspots. What is
completely contrary to the spirit of the modern synthesis is
targeted DNA
change that is
functionally significant. We now have examples of that
kind of change. To read more on those examples see the
answer to the question on the
relevance to physiology.

The relevant part of
the IUPS2013 lecture starts at 7 minutes with the phrase “It
is important to ask the question what we mean by random”
followed at 7:18 by “rather by whether the changes are
functionally relevant” before quoting the paragraph from the
article shown above. The article and the lecture could not
be clearer. See also Randomness and
Function.

It is remarkable how often the same unthinking criticism of
the article and the lecture turn up on blog websites. What
this shows is that the writers have not taken the time to
read the article or listen carefully to the video lectures.
Having missed the target on this matter, the same bloggers
usually go on to the further false accusation that I claim
to have disproved “the theory of evolution”. Anyone who
reads the article or watches the videos would find that
laughable, so why do they do it? For some strange reason
defenders of neo-Darwinism on blogs seem to think that
anyone who questions neo-Darwinism is questioning the
existence of evolution. The article clearly states:

“In some respects, my article
returns to a more nuanced, less dogmatic view of
evolutionary theory (see also Müller, 2007; Mesoudi et al.
2013), which is much more in keeping with the spirit of
Darwin’s own ideas than is the Neo-Darwinist view.”

As I write in another answer (Answers-dogmatism.html):
“It is perfectly possible to defend the virtual certainty
that life has evolved while debating in the usual
argumentative scientific way the uncertainties surrounding
the question of mechanisms.” That is where the real debate
is taking place: what are the relative contributions of the
various mechanisms to the evolutionary process and the speed
with which it has occurred (see
Speed of Evolution).